[Paper Review] Towards $A + B$ theory in conifold transitions for Calabi-Yau threefolds

Yuan‐Pin Lee, Hui-Wen Lin+1 more|arXiv (Cornell University)|Feb 11, 2015

Algebraic Geometry and Number Theory27 references4 citations

TL;DR

This paper establishes that the combined $A$-model (Gromov–Witten theory in all genera) and $B$-model (variation of Hodge structures) on a Calabi–Yau threefold $X$ undergoing a conifold transition to $Y$ determines the corresponding theories on $Y$, and vice versa, via a $ frac{1}{2}$-exact sequence linking vanishing cycles and exceptional curves. The key result is a $ frac{1}{2}$-duality between excess $A$ and $B$ theories, suggesting a unified $A+B$ theory through logarithmic connections on the moduli space.

ABSTRACT

For projective conifold transitions between Calabi-Yau threefolds $X$ and $Y$, with $X$ close to $Y$ in the moduli, we show that the combined information provided by the $A$ model (Gromov--Witten theory in all genera) and $B$ model (variation of Hodge structures) on $X$, linked along the vanishing cycles, determines the corresponding combined information on $Y$. Similar result holds in the reverse direction when linked with the exceptional curves.

Motivation & Objective

• To understand how Gromov–Witten theory ($A$-model) and variation of Hodge structures ($B$-model) transform under projective conifold transitions between Calabi–Yau threefolds.
• To establish a mutual determination between the $A$- and $B$-models on $X$ and $Y$ via the linking of vanishing cycles and exceptional curves.
• To construct a $ frac{1}{2}$-duality between excess $A$- and $B$-theories using logarithmic connections on the moduli space.
• To provide a framework for a unified $A+B$ theory in the context of conifold transitions.

Proposed method

• Use of the exact sequence $0 \to H^2(Y)/H^2(X) \xrightarrow{B} \mathbb{C}^k \xrightarrow{A^t} V \to 0$ to relate topological data of $X$ and $Y$ via vanishing cycles and exceptional curves.
• Construction of a $ frac{1}{2}$-duality via the logarithmic part of the Gauss–Manin connection on $V$ and the Dubrovin connection on $H^2(Y)/H^2(X)$.
• Application of the Gauss–Manin connection on a smooth family over a non-reduced base $\mathcal{Z}_1 = Z_{\mathcal{M}_{\bar{X}}}(\mathscr{I}^2)$ to extend the variation of Hodge structures to the boundary of the moduli space.
• Lifting the Hodge filtration $F^\bullet$ from $H^3(U,\mathbb{C})$ to $H^3(X,\mathbb{C})$ using the first Hodge–Riemann bilinear relation and Griffiths transversality.
• Use of the period map $\phi(r,s) = \exp(\sum_i \frac{\log w_i}{2\pi i} N^{(i)}) \psi(r,s)$ to reconstruct the $B$-model on $X$ from the $B$-model on $Y \setminus Z = U$, with monodromy determined by the relation matrix $A$.
• Proof that the period map $\phi$ is fully determined by the relation matrix $A$, hence the $B$-model on $X$ is determined by the $B$-model on $Y$.

Experimental results

Research questions

• RQ1 Can the $A$- and $B$-models on $X$ and $Y$ be mutually determined via conifold transitions?
• RQ2 How do the logarithmic parts of the Gauss–Manin and Dubrovin connections on the excess theories glue to form a trivial theory?
• RQ3 To what extent does the $B$-model on $Y \setminus Z$ determine the $B$-model on $X$?
• RQ4 How is the period map on $\mathcal{M}_X$ reconstructed from the geometry of $Y$ and the relation matrix $A$?
• RQ5 What is the role of the moduli space $\mathcal{M}_{\bar{X}}$ and its subvariety $\mathcal{M}_Y$ in encoding the $A+B$ structure?

Key findings

• The $A$-model on $X$ and the $B$-model on $Y$ are mutually determined via the exact sequence linking $H^2(Y)/H^2(X)$ and $V$, the space of vanishing cycles.
• The logarithmic part of the Gauss–Manin connection on $V$ and the Dubrovin connection on $H^2(Y)/H^2(X)$ arise from a trivial logarithmic connection on $(\underline{\mathbb{C}} \oplus \underline{\mathbb{C}}^\vee)^k$, indicating a $\tfrac{1}{2}$-duality between excess $A$- and $B$-theories.
• The period map $\phi: \mathcal{M}_X \to \mathbb{D}/\Gamma$ is completely determined by the relation matrix $A$ of the exceptional curves $C_i$, showing that the $B$-model on $X$ is encoded in the $B$-model on $Y \setminus Z$.
• The Hodge filtration $F^\bullet$ on $H^3(X,\mathbb{C})$ is uniquely lifted from $H^3(U,\mathbb{C})$ using the first Hodge–Riemann bilinear relation, ensuring the existence of a maximal horizontal slice to the period domain.
• The deformation space $\operatorname{Def}(Y)$ embeds into $H^1(U, T_U) \cong \operatorname{Def}(\bar{X})$, and $\mathcal{M}_Y$ is a submanifold of $\mathcal{M}_{\bar{X}}$, with $\mathcal{Z}_1$ being the first jet extension of $\mathcal{M}_Y$ in $\mathcal{M}_{\bar{X}}$.
• The $B$-model on $X$ is fully reconstructed from the $B$-model on $Y \setminus Z = U$ via the period map, which depends only on the matrix $A$, proving that the $B$-model on $X$ is determined by the $B$-model on $Y$.